Branching Processes, Random Partial Differential Equations and Applications

在异质和随机环境中的物理、生物和经济动力学系统在自然界中无处不在，包括生态学中的传播、经济增长和流体湍流等各种各样的例子。这类系统的数学建模涉及非线性偏微分方程和随机过程，对于现实世界的应用程序包含大量的时间和空间尺度，使得其行为的微观细节的数值模拟甚至超出了现代计算机的范围。为了缓解这个问题，一种方法是使用近似的宏观有效模型。另一种方法是考虑简化的模型，这些模型继承了更大系统动态的定性特性。该项目的总体目标是研究这种近似在物理科学和宏观经济学等方面的应用的有效性。项目的第一部分旨在开发新的工具，更好地理解分支过程和非线性抛物方程之间的联系。其中一个重点是研究分支布朗运动的维度高于一。第二部分关注流体动力学和反应扩散模型中产生的非线性方程的长时间行为，重点关注非常长的时间尺度，当解中的波动开始以超出经典中心极限定理时间尺度的方式建立时。第三部分研究宏观经济学中出现的平均场方程组的解的长期行为。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Physical, biological, and economic dynamics systems in heterogeneous and random environments are ubiquitous in nature, and include examples as diverse as spreading in ecology, economic growth, and fluid turbulence. The mathematical modeling of such systems involves nonlinear partial differential equations and stochastic processes, which for real-world applications contain a multitude of temporal and spatial scales, making the numerical simulation of the microscopic details of their behavior beyond reach even of modern computers. To mitigate this issue, one approach is to use approximate macroscopic effective models. Another is to consider simplified models that inherit the qualitative properties of the dynamics of the larger system. The overarching goal of this project is to study the validity of such approximations in applications of interest to physical sciences and macroeconomics, among others. The mentoring and training of graduate students will be integrated in the project.The first part of the project aims to develop new tools and better understanding of the connection between branching processes and nonlinear parabolic equations. One focus is on the study of branching Brownian motion in dimensions higher than one. The second part concerns the long-time behavior of nonlinear equations arising in the fluid dynamics and reaction-diffusion modeling, with a focus on very long-time scales, when fluctuations in the solutions start building up in ways beyond the classical central limit theorem time scales. The third part investigates the long-time behavior of the solutions to systems of mean-field equations that arise in macroeconomics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.